Understanding How SVD Helps in Movie Recommendation¶
We are given a dataset of user-movie ratings (R) that looks like the following:
Ratings in the matrix range from 1-5. In addition, the matrix contains nan wherever there is no rating provided by the user for the corresponding movie. One simple way to utilize this matrix to predict movie ratings for a given user-movie pair would be to fill in each row / column with the average rating for that row / column.
For example: For each movie, if any rating is missing, we could just fill in the average value of all available ratings and expect this to be around the actual / expected rating.
While this may sound like a good approximation, it turns out that by just using SVD we can improve the accuracy of the predicted rating.
How does SVD fit into this picture?¶
Recall how we previously used SVD to compress images by throwing out less important information. We could apply the same idea to our above matrix (R) to generate another matrix (R_) which will provide the same information, i.e ratings for any user-movie pairs but by combining only the most important features.
Let's look at this with an example:
Assume that decomposition of matrix R looks like:
$$
R = U\Sigma V^{T}
$$
We can re-write this decomposition as follows:
$$
R = U\sqrt\Sigma \sqrt\Sigma V^{T}
$$
If we were to take only the top K singular values from this matrix, we could again write this as:
$$
R\_ = U\sqrt\Sigma_k \sqrt\Sigma_k V^{T}
$$
Thus we have now effectively separated our ratings matrix R into two matrices given by:
$
U_k = U_{[:k]}\sqrt\Sigma_k
$
and
$
V_k = \sqrt\Sigma_k V_{[:k]}^{T}
$
There are many ways to visualize the importance of $U$ and $V$ matrices but with respect to our context of movie ratings, we can visualize these matrices as follows:
We can imagine each row of $U_k$ to be holding some information how much each user likes a particular feature (feature1, feature2, feature 3...feature k). On the contrary, we can imagine each column of $V_k^{T}$ to be holding some information about how much each movie relates to the given features (feature 1, feature 2, feature 3 ... feature k).
Lets denote the row of $U_k$ by $u_i$ and the column of $V_k^{T}$ by $m_j$. Then the dot-product: $u_i * m_j$ can provide us with information on how much a user i likes movie j.
What have we achieved by doing this?¶
Starting with a matrix R containing very few ratings, we have been able to summarize the sparse matrix of ratings into matrices $U_k$ and $V_k$ which each contain feature vectors about the Users and the Movies. Since these feature vectors are summarized from only the most important K features (by our SVD), we can predict any User-Movie rating that is closer to the actual value than just taking any average rating of a row / column (recall our brute force solution discussed above).
Now this method in practice is still not close to the state-of-the-art but for a naive and simple method we have used, we can still build some powerful visualizations as we will see in part 3.
We have divided the task into 3 parts:
1) Implement recommender_svd to return matrices $U_k$ and $V_k$
2) Implement predict to predict top 3 movies a given user would watch
3) (Ungraded) Feel free to run the final cell labeled to see some visualizations of the feature vectors you have generated
